Holmstrom B15 Atlas

Would like a big map of all relevant functors and cohomology theories.

Jakob’s motives diagram in thesis.

Schemes over Z, number rings, Q and number fields, local fields, p-adic integers, reals and complex numbers, adeles.

Set, Grp, Rings, R-mod, R-alg, Sch, Sch/S, Sm/S, PreShv(C), Shv(T), Top, Sset, Cat, various model cats, stacks(?), higher cats.

We should account for the fact that any cohomology theory defines a sheaf, and many other similar facts.

Should include all kinds of cohomology functors, including the abstract forms of algebraic K-theory (there are probably other unknown similar functors.)

Realization functors in motivic homotopy theory.

Weibel’s roadmap.

Show which things factor through various homotopy types.

An affine scheme is a by definition a representable covariant functor from Rings to Set.

FGL, $RPS_1$, $A_n$, $\hat{A}_n$, $R \mapsto R/nR$. K-theory of rings.

Localisation functors (also for rings and modules).

Picard group of a ring (through projective modules of rank one)

(Group ring with coefficients in a fixed ring.)

The nerve functor from Cat to sSet.

Representable functors. The Artin approximation thm is motivated by rep qs for functors on schemes.

Note: Bousfield and Kan LNM304 p 7 states that any functor from groups to groups can be “prolonged” to a functor from spaces to spaces. Example: R-completion.

An example: For “singular homology of varieties”, one constructs a complex $C(X)$. Taking the homology of $C(X) \otimes^L A$ gives singular homology with coeffs in $A$, and taking homology (?) of $RHom(C(X), A)$ gives singular cohomology with coeffs in $A$. This indicates a way of understanding the unification of geometric objects and coefficients.

One possible organizing principle is the following: Classify CTs according to what equivalence relation they induce on algebraic cycles. For example, motivic cohomology induces rational equivalence, while l-adic cohomology induces homological equivalence. This classification should give constraints on when there can exist maps from a theory to another. There is also one category of classical motives for each equivalence relation. A wild guess: Perhaps intersection cohomology induces numerical equivalence??? If not, is there a theory that does? What about algebraic equivalence? For algebraic eq, see Walker and Friedlander in K-theory handbook.

nLab page on B15 Atlas